ALGEBRAIC TOPOLOGY I

685 Pages·2017·9.68 MB·English

ALGEBRAIC TOPOLOGY I - II
STEFAN FRIEDL
Contents
References......................................................................... 6
1. Topological spaces............................................................. 15
1.1. The de(cid:12)nition of a topological space....................................... 15
1.2. Constructions of more topological spaces .................................. 25
1.3. Further examples of topological spaces..................................... 28
1.4. The two notions of connected topological spaces........................... 32
1.5. The (path-) components of a topological spaces............................ 35
1.6. Local properties........................................................... 37
1.7. Graphs and topological realizations of graphs.............................. 39
1.8. The basis of a topology.................................................... 43
1.9. Manifolds................................................................. 44
1.10. The classi(cid:12)cation of 1-dimensional manifolds............................. 48
1.11. Orientations of manifolds................................................. 52
2. Differential topology........................................................... 55
2.1. The Tubular Neighborhood Theorem...................................... 55
2.2. The connected sum operation ............................................. 59
2.3. Knots and their complements.............................................. 61
3. How can we show that two topological spaces are not homeomorphic? ......... 65
4. The fundamental group ....................................................... 69
4.1. Homotopy classes of paths................................................. 69
4.2. The fundamental group of a pointed topological space..................... 76
5. Categories and functors ....................................................... 82
5.1. De(cid:12)nition and examples of categories...................................... 82
5.2. Functors.................................................................. 84
5.3. The fundamental group as functor......................................... 86
6. Fundamental groups and coverings ............................................ 91
6.1. The cardinality of sets..................................................... 91
6.2. Covering spaces........................................................... 93
6.3. The lifting of paths........................................................105
6.4. The lifting of homotopies..................................................108
6.5. Group actions and fundamental groups.................................... 114
6.6. The fundamental group of the product of two topological spaces........... 120
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6.7. Applications: the Fundamental Theorem of Algebra and the Borsuk-Ulam
Theorem...............................................................123
7. Homotopy equivalent topological spaces ....................................... 126
7.1. Homotopic maps.......................................................... 126
7.2. The fundamental groups of homotopy equivalent topological spaces........ 128
7.3. The wedge of two topological spaces....................................... 133
8. Basics of group theory.........................................................140
8.1. Free abelian groups and (cid:12)nitely generated abelian groups.................. 140
8.2. The free product of groups................................................ 146
8.3. An alternative de(cid:12)nition of the free product of groups..................... 153
9. The Seifert-van Kampen theorem I............................................ 156
9.1. The Seifert{van Kampen theorem I........................................156
9.2. Proof of the Seifert-van Kampen Theorem 9.2............................. 163
9.3. More examples: surfaces and the connected sum of manifolds.............. 168
10. Presentations of groups and amalgamated products...........................174
10.1. Basic de(cid:12)nitions in group theory......................................... 174
10.2. Presentation of groups ................................................... 175
10.3. The abelianization of a group ............................................ 180
10.4. The amalgamated product of groups ..................................... 184
11. The general Seifert-van Kampen Theorem.................................... 189
11.1. The formulation of the general Seifert-van Kampen Theorem............. 189
11.2. The fundamental groups of surfaces...................................... 191
11.3. Non-orientable surfaces...................................................197
11.4. The classi(cid:12)cation of closed 2-dimensional (topological) manifolds......... 199
11.5. The classi(cid:12)cation of 2-dimensional (topological) manifolds with boundary.200
11.6. Retractions onto boundary components of 2-dimensional manifolds....... 205
12. Examples: knots and mapping tori........................................... 209
12.1. An excursion into knot theory ((cid:3))........................................ 209
12.2. Mapping tori.............................................................214
13. Limits ....................................................................... 224
13.1. Preordered and directed sets ............................................. 224
13.2. The direct limit of a direct system........................................225
13.3. Gluing formula for fundamental groups and HNN-extensions ((cid:3)).......... 237
13.4. The inverse limit of an inverse system.................................... 242
13.5. The pro(cid:12)nite completion of a group ((cid:3))...................................250
14. Decision problems............................................................252
15. The universal cover of topological spaces ..................................... 255
15.1. Local properties of topological spaces.....................................255
15.2. Lifting maps to coverings.................................................256
15.3. Existence of covering spaces.............................................. 261
16. Covering spaces and manifolds ............................................... 274
16.1. Covering spaces of manifolds............................................. 274
ALGEBRAIC TOPOLOGY I - II 3
16.2. The orientation cover of a non-orientable manifold........................278
17. Complex manifolds...........................................................281
18. Hyperbolic geometry......................................................... 287
18.1. Hyperbolic space......................................................... 287
18.2. Angles in Riemannian manifolds..........................................293
18.3. The distance metric of a Riemannian manifold ........................... 295
18.4. The hyperbolic distance function......................................... 298
18.5. Complete metric spaces.................................................. 300
19. The universal cover of surfaces ............................................... 302
19.1. Hyperbolic surfaces ...................................................... 302
19.2. More hyperbolic structures on the surfaces of genus g (cid:21) 2 ((cid:3)).............306
19.3. More examples of hyperbolic surfaces.....................................308
19.4. The universal cover of surfaces........................................... 313
19.5. Proof of Theorem 19.9 I..................................................314
19.6. Proof of Theorem 19.9 II................................................. 318
19.7. Picard’s Theorem........................................................ 321
20. The deck transformation group ((cid:3))........................................... 325
21. Related constructions in algebraic geometry and Galois theory ((cid:3))............ 336
21.1. The fundamental group of an algebraic variety ((cid:3))........................336
21.2. Galois theory ((cid:3)).........................................................338
22. CW-complexes............................................................... 340
22.1. De(cid:12)nition of (cid:12)nite-dimensional CW-complexes and examples ............. 340
22.2. Two topologies on R1 ................................................... 346
22.3. In(cid:12)nite-dimensional CW-complexes.......................................349
22.4. Properties of CW-complexes ............................................. 350
22.5. The Homotopy Extension Theorem.......................................359
22.6. Fundamental groups of CW-complexes ................................... 362
22.7. The Cellular Approximation Theorem.................................... 368
22.8. Proof of Proposition 22.20 ((cid:3))............................................ 371
22.9. Coverings of CW-complexes.............................................. 376
22.10. Spanning trees of graphs ((cid:3)) ............................................ 381
23. Higher homotopy groups..................................................... 383
23.1. De(cid:12)nition of the higher homotopy groups.................................383
23.2. Properties and calculations of the higher homotopy groups ............... 390
23.3. Covering spaces and higher homotopy groups.............................392
23.4. Are there any higher homotopy groups that are non-trivial?.............. 395
23.5. The Poincar(cid:19)e Conjecture................................................. 398
24. The homology groups of a topological space.................................. 402
24.1. Singular chains...........................................................402
24.2. De(cid:12)nition of the homology groups of a topological space..................405
24.3. First calculations of homology groups .................................... 410
24.4. Algebraic chain complexes................................................412
4 STEFAN FRIEDL
24.5. The functoriality of homology groups.....................................414
24.6. Direct products and direct sums ((cid:3))...................................... 415
24.7. The homology groups of a direct sum.....................................416
25. Homology and homotopies ................................................... 418
25.1. Chain homotopies........................................................ 418
25.2. Homology and homotopic maps .......................................... 419
26. Long exact sequences and the homology of quotient spaces ................... 426
26.1. Long exact sequences.....................................................426
26.2. The homology groups of spheres..........................................429
26.3. Basic homological algebra................................................ 432
26.4. Relative homology groups................................................ 438
26.5. The Excision Theorem................................................... 445
26.6. The proof of the Excision Theorem 26.16: the idea....................... 447
26.7. The proof of the Excision Theorem 26.16: the full details................. 448
26.8. Explicit generators of homology groups...................................461
26.9. Applications to topological manifolds.....................................467
27. The degree of a self-map of a sphere..........................................470
28. The Mayer{Vietoris sequence and its applications ............................ 478
28.1. Split exact sequences.....................................................478
28.2. The Mayer{Vietoris sequence.............................................481
28.3. Applications of the Mayer{Vietoris sequence..............................484
28.4. The Mayer{Vietoris Theorem for CW-complexes ......................... 487
28.5. The homology groups of the torus and the Klein bottle................... 488
28.6. The homology groups of a knot complement..............................493
28.7. The homology groups of a mapping torus ((cid:3))............................. 495
29. Cellular homology............................................................500
29.1. The homology groups of a nested sequence of topological spaces.......... 500
29.2. The de(cid:12)nition of cellular homology....................................... 503
29.3. The relationship between cellular and singular homology ................. 506
29.4. The cellular boundary maps..............................................511
29.5. The homology groups of 2-dimensional manifolds......................... 516
29.6. The local degree ......................................................... 520
30. The Jordan Curve Theorem.................................................. 530
31. Topological robotics..........................................................538
31.1. The matrix groups SO(3) and SU(2).....................................538
31.2. The belt trick............................................................ 543
31.3. Topological robotics......................................................545
31.4. Linkages................................................................. 546
32. The (cid:12)rst homology group and the fundamental group.........................550
32.1. The Hurewicz homomorphism............................................ 550
32.2. Natural transformations..................................................557
32.3. The Hurewicz homomorphism in higher dimensions.......................562
ALGEBRAIC TOPOLOGY I - II 5
33. Simplicial complexes and homology groups of manifolds ...................... 566
33.1. Simplicial complexes..................................................... 566
33.2. The top-dimensional homology group of a manifold.......................569
33.3. The fundamental class of a compact orientable manifold.................. 576
33.4. The homology groups of the connected sum of two manifolds............. 584
33.5. Representing homology classes by manifolds ((cid:3)).......................... 587
33.6. The degree of a map between oriented manifolds ......................... 589
34. The Euler characteristic...................................................... 594
34.1. The Euler characteristic and homology groups............................594
34.2. Properties of the Euler characteristic..................................... 597
34.3. The Euler characteristic of the product of two CW-complexes ............ 602
34.4. Groups acting on spheres.................................................604
34.5. Graphs ((cid:3))............................................................... 605
34.6. The Lefschetz Fixed Point Theorem ((cid:3)).................................. 606
35. Applications of the Euler characteristic ((cid:3))................................... 608
35.1. Building a leather football................................................608
35.2. Platonic solids ........................................................... 609
35.3. Homology spheres........................................................614
35.4. Planar graphs............................................................ 616
36. Homology with coefficients................................................... 621
36.1. The tensor product of abelian groups.....................................621
36.2. The tensor product of a chain complex with an abelian group ............ 626
36.3. Exact functors........................................................... 631
36.4. The G-torsion of an abelian group........................................634
36.5. The Universal Coefficient Theorem....................................... 644
36.6. Splittings of the Universal Coefficient Theorem........................... 649
36.7. Isomorphisms on homology groups induce chain homotopies ((cid:3))...........652
36.8. Homological algebra over an arbitrary commutative ring ((cid:3)).............. 654
37. The Ku(cid:127)nneth Theorem....................................................... 656
37.1. The tensor product of chain complexes................................... 656
37.2. The Eilenberg-Zilber Theorem............................................657
37.3. The Ku(cid:127)nneth Theorem for chain complexes...............................661
37.4. The Ku(cid:127)nneth Theorem for topological spaces.............................664
38. Applications of homology groups............................................. 668
38.1. Persistent homology ((cid:3)).................................................. 668
38.2. Division algebras.........................................................669
38.3. The transfer map ........................................................ 675
38.4. The Borsuk-Ulam Theorem and the Ham-Sandwich Theorem.............679
6 STEFAN FRIEDL
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